Phase steps: Nonparametric extensions to inspiral template families

1
Phase stepsNonparametric extensions to
inspiral template families

• GWDAW-9
• Dec-16-2004
• R. OShaughnessy, D. Jones
• Northwestern University

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Context Motivation

• Pipelines
• NS-NS PN
• BH-BH BCV
• BH-NS BCVspin

good (!)
3
Context Motivation

• Pipelines
• NS-NS PN
• BH-BH BCV
• BH-NS BCVspin

less physical (detection only) more
templates, higher threshold
4
Motivation

• Present pipelines good !
• Will find what were looking for, if there

)? see owen latest
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Context Motivation

• Potential improvements?
• Faster searching
• Inspiral search is very CPU-intensive
• but still need (eventually) to do real-time
• Can we get comparable (or more) coverage w/ a
  fast search?
• (e.g., as w/ BCVspin -- fast search over
  extrinsic parameters)

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Context Motivation

• Potential improvements?
• Faster searching
• Greater Coverage
• No guarantees about physical signal FF for
  example
• BCV fits just what weve tried so far (PN, EOB),
  but
• Merger waves?
• Non-GR - detecting signals from non-GR theories
• Try models w/ more template parameters?
• More coverage
• Only slow increase in threshold SNR with template
  number
• Problem computational cost ? ? w/ more
  templates / intrinsic parameters
• but adding a few extrinsic parameters ok
• e.g. BCVspin

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Context Motivation

• Potential improvements?
• Faster searching
• Greater Coverage
• Our proposal
• Adds parameters to conventional families
• Many more templates .
• but still fast search -- effectively extrinsic
• Result just good for detection (not estimation)
• Good coverage
• ..requires careful tuning to get desired false
  rate

very preliminary
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Analysis Assumptions

• Simple toy proposal analysis
• Concreteness Restrict study to
• LIGO 1 noise . gaussian only
• BH-NS mass ranges
• 0 PN amplitude (i.e. ?0)
• Extending nonspinning PN templates

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Outline

• Context Motivation why?
• Proposed form what?
• Phase steps why this form?
• How much overlap improvement
• Complications works?
• False alarms with steps
• Conclusions

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Choosing a form

• Examine residuals?
• Expected residuals after template maximization?
• try to choose additions to compensate
• Concretely
• Phase residuals only

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Choosing a form

• Examine phase residuals?
• Monotonic phase variation?
• well-fit by templates --gt irrelevant
• Sinusoidal phase errors
• except maximization over extrinsic tc
  automatically reshapes the phase error curve
• Example
• Adjoin artificial sinusoidal phase error
• Maximize tc, phic

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Choosing a form

• Examine phase residuals?
• Smooth phase variation --gt irrelevant
• Sinusoidal phase errors --gt reshaped
• Example
• Artificial sinusoidal phase error

??
before
Hz
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Choosing a form

• Examine phase residuals?
• Smooth phase variation --gt irrelevant
• Sinusoidal phase errors --gt reshaped
• Example
• Artificial sinusoidal phase error
• Maximization requires phase constant near max
  sensitivity

??
d?/df
after (FF0.64)
Hz
Hz
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Choosing a form

• Examine phase residuals?
• Smooth phase variation --gt irrelevant
• Sinusoidal phase errors --gt reshaped
• residuals suggest
• use (smoothed) step functions in phase

??
d?/df
after (FF0.64)
Hz
Hz
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Phase steps

• Motivation
• Add to ? smooth changes 0-gt2 ?
• Equations

local effect on overlap
d?/df
Hz
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Phase steps Example

• Start
• Sinusoidal phase
• Example (above)
• Optimize
• Fix tc, phic
• Add one by one
• Optimize each spike
• independently
• Results
• Initially overlap 0.64
• First step Fc50, w8.2 overlap 0.74
• Second step Fc250, w8.5 overlap 0.70
• Both overlap 0.84

Hz
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Complications?

• Infinite-dimensional manifold
• suggests slow search
• but local character of steps --gt fast (rough)
  search
• Easy to excite steps by noise
• true, but manageable via search constraints (
  rest of talk)

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Controlling False Alarms

• Familiar Question
• Improved overlap --gt improved detection rate
• More models --gt higher false alarm rate
• e.g., want detection threshold for fixed false
  rate
• ? goal Tailor parameter ranges to minimize
  increase in false rate
• How to avoid accidents
• Steps disjoint
• Steps arent too narrow or irrelevant
• Each step makes a significant change in SNR
• Look for steps only above a minimum SNR cutoff

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Controlling False Alarms

• How to avoid accidents
• Steps disjoint
• Sanity constraint
• Steps of opposite sign
• cancel
• (1) (-1) 0 in same place
• Steps of same sign
• narrower than w
• (I.e. d?/df larger)
• --gt annoying

?
Require ( f1-f2 )gt 6(w1w2) for any pair of
steps
Hz
Side effect z is additive
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Controlling False Alarms

• How to avoid accidents
• Steps disjoint
• Steps arent too narrow or irrelevant
• Certain steps cant change SNR much
• Narrow steps (w ltlt 1)
• Steps centered away from maximum
• by definition, small-z steps
• However, easy to excite
• SNR in band of fraction z determines excitation
• P(error)

Interlude SNR band-limited issues
Net coherent amplitude
Net noise power
z-band amplitude
z-band noise power
relative bandwidth
Require zgt0.1
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Controlling False Alarms

• How to avoid accidents
• Steps disjoint
• Steps arent too narrow or irrelevant
• Certain steps cant change SNR much
• Narrow steps (w ltlt 1)
• Steps centered away from maximum
• by definition, small-z steps
• However, easy to excite
• SNR in band of fraction z determines excitation
• P(error)

Z measures 1)
characteristic ??/? due to step if
indeed present 2) effective width of band
(proportional to relative,
power-weighted width)
Interlude SNR and steps band-limited issues
relative bandwidth
Require zgt0.1
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Controlling False Alarms

• How to avoid accidents
• Steps disjoint
• Steps arent too narrow or irrelevant
• Small-z steps
• Narrow steps (wltlt1), or
• Far from sensitivity maximum
• False positives with given step?
• Probability of false narrow step
• probability N(0,1)
• gt SNR-in-band
• can be large if z small !

• Models
• Signal
• SNR ?
• Two hypotheses (templates)
• Signal
• Signal (one fixed step _at_ z)

Require zgt0.1
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Controlling False Alarms

• How to avoid accidents
• Steps disjoint
• Steps arent too narrow or irrelevant
• Step makes a significant change in SNR
• Band-limited SNR
• Total SNR change
• Band-limited SNR higher
• because occurs only in smaller band
• Accidents unlikely, but still possible

Require
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Controlling False Alarms

• How to avoid accidents
• Steps disjoint
• Steps arent too narrow or irrelevant
• Each step makes a significant change in SNR
• Look for steps only above a minimum SNR cutoff
• Require
• Dont start looking unless SNR already high
• ? gt?s
• (e.g, close to detection threshold ?s ?d --
  ?? )
• Reason
• Two moderately unlikely accidents required to
  find spike
• SNR gt ?s
• d(SNR) gt 0.9

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• Aside Nonparametric perspective
• Use more templates when SNR warrants
• implicitly / automatically implemented by this
  constraint
• (INTERLUDE)

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Controlling False Alarms

• How to avoid accidents
• Steps disjoint
• Steps arent too narrow or irrelevant
• Each step makes a significant change in SNR
• Look for steps only above a minimum SNR cutoff
• Detection strategy
• Conventional intrinsic search at each set of
  extrinsic params
• Trial step search
• Dont try steps unless starting SNR gt threshold
  ?s
• Each step must improve SNR by at least ??s
• Search over extrinsic parameters
• Detection if final SNR gt threshold ?d

Toy s ?s 7 ??s 1 ?d 8
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Summary Conclusions

• Status
• Generalizes any templates to nonparametric
  detection family
• Phase steps fix gross phase errors
• Family features match fine structure (e.g. spin)
• Pipeline (one-detector)
• Probes into noise below detection threshold of
  base family
• Detects signals which would otherwise be rejected
• (because our templates only imperfectly model the
  true signal)
• and which should not have happened by accident.
• Applications merger waves, testing whether GR
  describes signals, etc
• Further tests
• Coverage of full pipeline , w/ constraints vs
  SNR of input
• Real noise

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Nonparametric ?

• Templates used implicitly depend on SNR (!)
• Broad coverage for high SNR
• Low coverage for low SNR
• Pipeline as described
• not nonparametric (yet)